COMP9318 (18S1) ASSIGNMENT 1

Q1. (40 marks)

Consider the following base cuboid Sales with four tuples and the aggregate function

SUM:

Location T ime Item Quantity

Sydney 2005 PS2 1400

Sydney 2006 PS2 1500

Sydney 2006 Wii 500

Melbourne 2005 XBox 360 1700

Location, T ime, and Item are dimensions and Quantity is the measure. Suppose the

system has built-in support for the value ALL.

(1) List the tuples in the complete data cube of R in a tabular form with 4 attributes,

i.e., Location, T ime, Item, SUM(Quantity)?

(2) Write down an equivalent SQL statement that computes the same result (i.e., the

cube). You can only use standard SQL constructs, i.e., no CUBE BY clause.

(3) Consider the following ice-berg cube query:

SELECT Location, Time, Item, SUM(Quantity)

FROM Sales

CUBE BY Location, Time, Item

HAVING COUNT(*) > 1

Draw the result of the query in a tabular form.

(4) Assume that we adopt a MOLAP architecture to store the full data cube of R, with

the following mapping functions:

fLocation(x) =

1 if x = ‘Sydney’,

2 if x = ‘Melbourne’,

0 if x = ALL.

fT ime(x) =

1 if x = 2005,

2 if x = 2006,

0 if x = ALL.

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2 DUE ON 23:59 23 MAY, 2018 (WED)

fItem(x) =

1 if x = ‘PS2’,

2 if x = ‘XBox 360’,

3 if x = ‘Wii’,

0 if x = ALL.

Draw the MOLAP cube (i.e., sparse multi-dimensional array) in a tabular form

of (ArrayIndex, V alue). You also need to write down the function you chose to

map a multi-dimensional point to a one-dimensioinal point.

Q2. (30 marks)

Consider binary classification where the class attribute y takes two values: 0 or 1. Let the

feature vector for a test instance be a d-dimension column vector ~x. A linear classifier with

the model parameter w (which is a d-dimension column vector) is the following function:

y =

(

1 , if w>x > 0

0 , otherwise.

We make additional simplifying assumptions: x is a binary vector (i.e., each dimension

of x take only two values: 0 or 1).

• Prove that if the feature vectors are d-dimension, then a Na¨ıve Bayes classifier is

a linear classifier in a d + 1-dimension space. You need to explicitly write out the

vector w that the Na¨ıve Bayes classifier learns.

• It is obvious that the Logistic Regression classifier learned on the same training

dataset as the Na¨ıve Bayes is also a linear classifier in the same d + 1-dimension

space. Let the parameter w learned by the two classifiers be wLR and wNB, respectively. Briefly explain why learning wNB is much easier than learning wLR.

log Hint 1. Q

i =i x

P

i

i x log

Q3. (30 marks)

Consider a dataset consisting of n training data xi and the corresponding class label

yi ∈ { 0, 1 }.

(1) Consider the standard logistic regression model:

P[y = 1 | x] = σ(w>x)

where σ is the sigmoid function.

The learning of the model parameter is to find w∗

that minimizes some function

of w, commonly known as the loss function.

COMP9318 (18S1) ASSIGNMENT 1 3

Prove that the loss function for logistic regression is:

`(w) = Xn

i=1

−yiw>xi + ln(1 + exp(w>xi))

(2) Consider a variant of the logistic regression model:

P[y = 1 | x] = f(w>x)

where f : < → [0, 1] is a squashing function that maps a real value to a value

between 0 and 1.

Write out its loss function.

Submission

Please write down your answers in a file named ass1.pdf. You must write down

your name and student ID on the first page.

You can submit your file by

give cs9318 ass1 ass1.pdf

Late Penalty. -10% per day for the first two days, and -20% for each of the following

days.